Anomalously large spin-dependent electron correlation in the nearly half-metallic ferromagnet ${\mathrm{CoS}}_2$

The spin-dependent band structure of ${\mathrm{CoS}}_2$, which is a candidate for a half-metallic ferromagnet, was investigated by both spin- and angle-resolved photoemission spectroscopy and theoretical calculations to reappraise the half-metallicity and electronic correlations. We determined the three-dimensional Fermi surface and the spin-dependent band structure. As a result, we found that a part of the minority spin bands is on the occupied side in the vicinity of the Fermi level, providing spectroscopic evidence that ${\mathrm{CoS}}_2$ is not a half-metal but very close. Band calculations using density functional theory with generalized gradient approximation showed good agreement with the observed majority spin $e_g$ bands, while it could not explain the observed band width of the minority-spin $e_g$ bands. On the other hand, theoretical calculations using dynamical mean field theory could better reproduce the strong mass renormalization in the minority-spin $e_g$ bands. Our results strongly suggest the presence of anomalously enhanced spin-dependent electron correlation effects on the electronic structure in the vicinity of the half-metallic state. We also report the temperature dependence of the electronic structure across the Curie temperature and discuss the mechanism of the thermal demagnetization. Our discovery of the anomalously large spin-dependent electronic correlations not only demonstrates a key factor in understanding the electronic structure of half-metals but also provides a motivation to improve theoretical calculations on spin-polarized strongly correlated systems.

Figure 1

(a) Low-energy electron diffraction (LEED) pattern at an incident electron energy of 185 eV. (b) Brillouin zone of pyrite-type ${\mathrm{CoS}}_2$. (c) and (d) Angle-resolved photoemission spectroscopy (ARPES) intensity maps of the Fermi surface (FS) measured at $h\nu =66$ and 87 eV [corresponding to cuts P1 and P2 marked in (b)], respectively, overlaid with the fitted momentum distribution curve (MDC) peak positions at $E=E_F$ showing the $k_F$ positions (black circles). (e) and (f) $k_F$ plots along P1 and P2, respectively, symmetrized by assuming a twofold symmetry with respect to the Γ point. The thick shaded lines are guides for the eye.

Figure 2

(a) Angle-resolved photoemission spectroscopy (ARPES) intensity map as a function of energy up to $E_F$ and wave vector. (b) Second derivative of ARPES intensity map in (a) with respect to energy. (c) Polarization-dependent ARPES intensity map around the $X$ point: (left) circularly polarized light, (middle) $s$-polarized light, and (right) $p$-polarized light. (d) Second derivatives of ARPES intensity maps in (c) with respect to wave vector. Yellow filled circles show momentum distribution curve (MDC) peak positions estimated from the right panel in (c). (e) and (f) ARPES intensity maps along the $M\mathrm{\Gamma }X$ and the $R{X}^{\prime \prime }M$ lines, respectively, overlaid with plots showing peak positions of energy distribution curves (EDCs; empty circles) and momentum distribution curves (MDCs; filled circles). Yellow filled circles are the same as in (d) but folded with respect to the $X$ point.

Figure 3

(a) Two-dimensional Brillouin zone (gray lines) showing estimated measured $k$ positions for the states near $E_F$ (red curves) for each photon energy. (b) Angle-resolved photoemission spectroscopy (ARPES) intensity map of the Fermi surface (FS) as a function of $k_x$ and $k_y$ measured by 7 eV laser. The FS is obtained with an energy-integration window of ±1meV. (c) ARPES and (d) spin- and angle-resolved photoemission spectroscopy (SARPES) majority spin bands and (e) SARPES minority spin bands intensity maps as a function of energy to $E_F$ and $k_x$. (f)–(h) Same as (c)–(e) but measured in the vicinity of $E_F$. (i) and (j) Spin-resolved energy distribution curves (EDCs) at $k$ points corresponding to the blue shaded line cuts in (c) and (f). Black solid makers in (i) indicate peak positions of the structure around −1 eV in the majority and minority spin spectra.

Figure 4

Electronic structure of ${\mathrm{CoS}}_2$ in ferromagnetic (FM) state determined with generalized gradient approximation (GGA) + spin-orbit coupling (SOC) exchange correlation functional. (a) Band structure calculated with exchange splitting reduced to 74%. Both majority and minority bands are crossing $E_F$. (b) Corresponding Fermi surface (FS) with majority and minority FS sheets. (c) Experimentally obtained $k_F$ plots, same as Fig. 1. The thick shaded lines are guides for the eye. (d) Theoretical contour plots at $E-E_{\mathrm{F}}=0\mathrm{meV}$ (blue; corresponding to FS), $E-E_{\mathrm{F}}=10\mathrm{meV}$ (green), $E-E_{\mathrm{F}}=20\mathrm{meV}$ (yellow), and $E-E_{\mathrm{F}}=30\mathrm{meV}$ (red), calculated within GGA.

Figure 5

Comparison of our experimental angle-resolved photoemission spectroscopy (ARPES) results with band structures calculated within generalized gradient approximation (GGA). (a) ARPES intensity map taken by synchrotron light along the $\mathrm{\Gamma }X$ direction with peak positions of energy distribution curves (EDCs; blue empty circles) and momentum distribution curves (MDCs; blue and yellow filled circles), same as Fig. 1. (b) Band structure along the $\mathrm{\Gamma }X$ direction obtained from GGA + spin-orbit coupling (SOC) calculations with exchange splitting reduced to 74%. Red and blue colors indicate the projected majority and minority spin character, respectively. (c) Same as (a) but taken by 7 eV laser. (d) Same as (b) but enlarged near $E_F$. (e) Same as (a) but taken along the $MR$ direction. The light blue dotted line is the curve of the ζ band fitted with a cosine function to estimate the bandwidth. (f) Same as (b) but for the $MR$ direction.

Figure 6

Magnetic moment $M$ (red squares) and inverse of the ferromagnetic susceptibility 1/χ (blue squares) as a function of temperature, obtained dynamical mean-field theory (DMFT) calculations for $U=4.0\mathrm{eV}, J=1.0\mathrm{eV}$. $T_{\mathrm{C}}$ was determined from a linear fit of 1/χ by the condition $1/\chi =0$.

Figure 7

The single-particle excitation spectrum for $T/T_{\mathrm{C}}=0.58, U=4\mathrm{eV}$, and $J=1.0\mathrm{eV}$ for (a) the majority spin and (b) the minority spin calculated within density functional theory (DFT) + dynamical mean-field theory (DMFT). Momentum dependence of the zero-energy spectral functions in the $\mathrm{\Gamma }XM{X}^{\prime }$ plane in a 20 meV energy window for (c) the majority spin and (d) the minority spin calculated within DFT + DMFT and (e) obtained by angle-resolved photoemission spectroscopy (ARPES). (f)–(h) Same as (c)–(e) but in the $X^{\prime }MRM$ plane.

Figure 8

Comparison of our angle-resolved photoemission spectroscopy (ARPES) results with the spectral function maxima calculated within density functional theory (DFT) + dynamical mean-field theory (DMFT). Blue empty (filled) circles show the peak positions of energy distribution curves (EDCs) [momentum distribution curves (MDCs)]. Light blue dotted line is the curve of the ζ band fitted with a cosine function. Solid green lines show the peak position of the single-particle excitation spectra calculated within DFT + DMFT, which are shifted so that the energy at the $R$ point matches the experimental value.

Figure 9

(a) Temperature dependence of the angle-resolved photoemission spectroscopy (ARPES) intensity map along the $\mathrm{\Gamma }X$ line measured by 7 eV laser. (b) and (c) Energy distribution curves (EDCs) at a $k_x$ point marked by the dotted line in (a) and those divided by the Fermi function at measured temperatures which is convoluted with a 6 meV Gaussian corresponding to the energy resolution of the measurement. The EDCs are normalized by their intensity at −150 meV. (d) EDCs at a $k_x$ point marked by the dotted line in (a). The EDCs are normalized by the peak height at 1.0 eV. Black thick markers show positions of the structure corresponding to the α band. (e) Temperature evolution of the width of the spectral peak at $E_F$ ($\mathrm{\Delta }E$) shown in (c). The $\mathrm{\Delta }E$ plots are obtained after subtracting the energy resolution of our measurement, 6 meV. The gray dotted line denotes a fit of $\mathrm{\Delta }E=A+B{T}^2$, where $A$ and $B$ are fitting parameters, and $T$ is the temperature, to the data in a temperature range of 20–120 K. (f) Temperature evolution of the lifetime (τ) of the minority spin electrons occupying the δ band, estimated by $\tau =\hslash /\mathrm{\Delta }\mathrm{E}$.